A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Tuesday, April 28, 2009

A handful of homework sets, a few exams (most of them take-home), a couple dozen presentations.

There's not much left to do, not much left to say.

We're here now.

It seems like only now some of the lessons I've tried to teach all semester have begun to sink in.

"If you write what you know and what you need, you're halfway...if not further...to a full and valid proof."

I don't just say that to hear it be said. I say it because it's true.

And I think they're finally starting to believe me.

What else? Open my mouth, what other jewels fall out?

"You get out of a class more or less what you put into it."

"The folks who do well on the homework sets are the ones who get started on it right away."

Do I sound a million years old, or what? Who am I to speak?

Was there ever a time when I didn't know how to do a derivative, let alone compute a Galois group? And it's hard to remember that induction isn't a naturally occurring process, that we have to train our brains to mold themselves around its artificial angles and uncomfortable corners.

"...By inductive hypothesis we know that Γ(n+1)=n!..."

By now, nothing could be clearer.

I wonder if those same students who nodded casual assent as we plowed through today's inductive proof would recognize the confusion they themselves faced just two months back when first confronted with that arcane mathematical method. The way's been long and sinuous, and even knowing where it twists and turns it's easy to fall off to its side.

Who knows what takes us to where we are now in our lives?

I used to like to exercise my mind in what I called the "Causality Game": think about a major life event you've undergone, and trace it back, step by step, to its earliest observable genesis: n-1 begat n, n-2 begat n-1, n-3 begat n-2, and so forth and so on and so forth...at a certain point the exercise is absurd, a diagnosis of butterfly burps and other unanalyzable initial conditions.

In the middle of a panel discussion at the WAC/CAC Symposium at UNC Greensboro a week and a half ago I caught myself playing the Causality Game, if only for a minute or so. "What in the world has brought me here?" I asked myself. I was sitting in the center of a room filled mostly with rhetoricians and composition specialists, the people in front of me backed by a graceful arc of windows filled with bright blue sky and overlooking a lush and verdant campusscape. It was a gorgeous day outside.

I took mental stock.

"I'm 34 years old," I thought, "and I'm a mathematician. I feel strongly about more than math: writing matters, too. I care enough about writing to be here, to be in this room. In fact I'm surrounded by teachers of writing, and they're talking about writing. They're talking about assessing writing. Someone's just asked a question. What did she ask?"

It wasn't a question, it was a comment. It was a divisive comment, meant to be divisive. For an hour or more (before lunch, at which time the combatants no doubt held parley) it was Duke and Wake and Davidson versus all the rest of the room. Old habits die hard.

"What brought me here? What right have I?" I smiled to myself, I almost laughed out loud.

For a minute or so I clumsily fumbled with the knot that tied today to yesterday, trying to make sense of the tangled mess of words and numbers that sat upon my lap, and after a bit I said "ah fuck it" and let it drop.

Oughtn't I be happy enough knowing that I'm happy enough, and leave it at that? Who cares how I got here?

I watched the first episode of Carl Sagan's Cosmos yesterday on Hulu. It's been years since I've seen that, and seeing it again reminded me of just how strong an influence it had on me when I was a child. Sagan was a boyhood hero, and I can safely say that few individuals more strongly than he made me want to be a scientist of some kind. (My dad's about the only one that's got him beat.)

I just looked it up: Carl Sagan's been dead for over 12 years now. I'm glad that I had a chance to see him speak before he passed away.

Just that once.

He was as witty and wise as I knew he'd be.

And now he's gone. And now I'm here. And I've got now what I'd wanted then.

As I'm fond of saying, I'm blessed (there, I said it!) in that I get paid to do something I'm damned good at and that I love doing, so much so that I'd do it even if I didn't get paid. (This is good, because as of today I'm getting paid 0.5% less than last year to do what I do.)

What more could I want?

The wants are never-ending.

I want a roomful of peers and pupils who are in love with learning and who aren't afraid to share that love with each other.

I want unlimited time with which to elaborate every beautiful idea those folks can come up with.

I want unfettered, unfeigned, unrestricted, unlimited inquiry.

I want every who what when where why and how to be asked and answered.

I want to not be so goddamned tired when there's so much more to do, and I want the same for everyone around me.

I want...

...I want...

...I want this semester to be over.

Summer's nearly here, and as my wonderful friend Bedelia hinted on her Facebook page a few days back, next semester's already calling on us to atone for this term's teacherly sins with a drawn-out dunk in the salvific waters of early classroom preparation.

Saturday, April 25, 2009

What follows is a shadow of the proceedings of the Monday, March 30th trial reenactment, Newton v. Leibniz, performed by my Calculus I class. There is paraphrasis, but I've tried to preserve the most important passages as perfectly as I can. I know that I managed to catch several of the juiciest lines verbatim.

***

9:05 -- 9:07. Leibniz's lead attorney (played by Kent) makes his opening argument. "We plead with you that you look past Newton's fame and popularity, and merely focus on the facts."

9:07 -- 9:08. Newton's lead attorney (played by Silas) makes his opening argument. "We will prove our case through an analysis of our client's paper on infinite series, and we will call John Collins and Henry Oldenburg as witnesses attesting to his character."

9:08 -- 9:32. Leibniz's team mounts a defense. The first witness on behalf of Dr. Leibniz is Ehrenfried Tschirnhaus (played ably by Omar). He testifies to the upstanding character of his friend, and to his mathematical talent.

Tschirnhaus: "I knew him well. We met in 1665, and he visited in 1666. I worked with him further, on mathematical projects. He assisted me in my work on catacaustic curves (involving a geometric form of calculus) and natural philosophy. He was a profoundly good mathematician. He was also an honest man. Though he may have been given the opportunity to plagiarize, he didn't take advantage of this opportunity."

Newton's attorneys have nothing to ask in cross-examination.

The second witness called to the stand is a noted historical and mathematical expert (played by Olaf), who testifies as to the differences between the two scholars' mathematics works.

Leibniz's counsel: "So Newton and Leibniz used different techniques?"

Witness: "Yes."

Leibniz's counsel: "Is it likely that their methods were developed separately?"

Witness: "Yes."

Newton's counsel [in cross-examination]: "What, specificallyt, leads you to believe that the methods are different?"

Witness: "Though Newton's work [on calculus] was performed in the years 1665 and 1666, he didn't publish his work until after [Robert] Hooke died. Leibniz had no exposure to Newton in that time."

Newton's counsel [after a good deal of conversation with ihs client and colleague]: "If Leibniz had seen a manuscript of Newton's, is it reasonable that he would have stolen Newton's work?"

Witness: "Yes, reasonable. He did see Newton's work, though it didn't relate to calculus."

Newton's counsel: "Did Leibniz see those papers? Did he use them?"

Witness: "I don't know for certain."

The third witness called in Leibniz's defense is Leibniz him(her)self (played impeccably by Francine).

Leibniz's counsel [holding Exhibit A, a diagram of Leibniz's derivation of infinitesimals]: "Is this involved in your derivation of calculus?"

Leibniz: "Yes."

Leibniz's counsel: "Can you explain it, please?"

Leibniz: "This illustrates my formulation of calculus through the use of infinitesimals."

Leibniz's counsel: "When did you perform this work?"

Leibniz: "The work was completed by 1675."

Leibniz's counsel: "Is it true that you were in London around that time?"

Leibniz: "I was indeed visiting John Collins in 1676. At that point he showed me a copy of Newton's De Analysi, but I didn't take any notes on it."

Leibniz's counsel: "Why were you holding off on publishing your own work?"

Leibniz: "The Holy Roman Empire was a tricky place to publish at that time. Catholicism was on the outs, and I didn't want to do anything to draw attention to myself."

Newton's counsel now begins their cross.

Newton's counsel: "The word 'infinitesimal' appeared in Newton's work in 1665, did it not? Are we to believe that you got no information from Newton's manuscript?"

Leibniz: "I can't prove that I hadn't stolen from Newton, but I ask that you take me on my word that I did not do so. Infinitesimals are integral [no pun intended] to calculus, so the word should appear. On the other hand, I came up with the word 'calculus.' Moreover, many of our terms are different, like 'fluids' and 'fluxions.' After all, I'm not going to call it a duck if it's an infinitesimal triangle."

Newton's counsel: "Would there have been time for you to have found a new route to calculus, given the time delay between your reading my client's work and your publication of your own? Or maybe you changed the date on some of your manuscripts?"

Leibniz: "Why would I do that? Absolutely not."

Newton's counsel: "Could you have, though?"

Leibniz: "I don't date all of my notes, so I can't be sure of when they were created."

Leibniz's attorneys call their final witness, Sir Isaac Newton himself (played by Knut, who shows outstanding mastery of Newton's work on calculus).

Leibniz's attorney: "The Presidency of the Royal Society is a position of power, correct?"

Newton: "Yes. I had excessive power in that regard."

9:32 -- 9:38. The court recesses for a brief break. After the break Newton's side mounts an offensive.

9:38 -- 10:02. Newton's attorneys present their case. The first witness called to the stand is John Collins (played astutely by Mary Ellen).

Newton's counsel: "You and Sir Isaac Newton are colleagues, correct?"

Collins: "Yes, since 1670."

Newton's counsel: "Can you describe Sir Isaac?"

Collins: "He's peculiar, but brilliant. He has a sensitive soul. He's very averse to criticism and often withdraws into depression."

Newton's counsel: "Would you call him vindictive?"

Collins: "No."

Newton's counsel: "Is he honest?"

Collins: "Yes."

Newton's counsel: "Perhaps the most honest person you've known?"

Collins: "Not more honest than my mother, but Newton's definitely up there."

Leibniz's attorneys begin cross-examination.

Leibniz's counsel: "You were personally involved in Newton's promotion to the Presidency of the Royal Society, correct?"

Collins: "Yes. I voted for him."

Leibniz's counsel: "And you were partly responsible for introducing Leibniz to Newton's work?"

Collins: "Yes, I got them together. I had no idea that Leibniz would plagiarize Newton, though."

Leibniz's counsel: "You're positive that Leibniz plagiarized?"

Collins: "Yes. We have the letters proving it."

Leibniz's counsel: "Have you seen Leibniz's techniques?"

Collins: "I've seen his methods, and some of them look different, but that doesn't take away from the fact that he saw Newton's work."

Leibniz's counsel: "Are you aware of the publication dates that show my client's work appeared before Newton's?"

Collins: "It was publicly distributed work, even if it wasn't formally published."

Leibniz's counsel: "How well do you know Leibniz? How would you describe him?"

Collins: "He's brilliant."

Leibniz's counsel: "Brilliant enough to come up with calculus on his own?"

Collins: "Yes."

Leibniz's counsel: "If he could develop calculus on his own, why steal it from Newton?"

Collins: "That's a good question!"

The next witness to take the stand on behalf of Newton is Henry Oldenburg (played by Kevin).

Newton's counsel: "What is your relationship with Newton?"

Oldenburg: "I'm the Secretary of the Royal Society. I've been in frequent correspondence with him, and have had many personal interactions with him."

Newton's counsel: "What is Sir Isaac's character?"

Oldenburg: "He's easily discouraged by criticism from other people. He's never satisfied with his method, and he always tweaks his experiments over and over to make sure he's got it right."

Newton's counsel: "I read about that. When he was working on On Optiks, he deformed his own eye to learn how it would effect his vision." [At this point, for the only time in the trial, Silas broke character: "No kidding. I read that. It was insane."]

Oldenburg: "I pushed Newton to publish. I talked with him often as a friend and not as a scientist. He was kind of withdrawn. At one point he dropped out of correspondence for 19 months. But I personally saw his work develop in his letters."

Newton's counsel: "When was this?"

Oldenburg: "In the early 1670s, I think."

Newton's counsel: "Was the manuscript available by 1675?"

Oldenburg: "Yes."

Newton's counsel: "Would it have been available to Leibniz?"

Oldenburg: "Well, he was elected a member of the Royal Society in 1670."

Newton's counsel: "Was he a big name at that time?"

Oldenburg: "Yes."

Newton's counsel: "As big, intelligent, important as he was, does that mean he wouldn't plagiarize?"

Newton's counsel [on redirect]: "The issue, Mister Oldenburg, is not that Newton shared his work, but rather than Leibniz plagiarized it, correct?"

Oldenburg: "Correct."

Newton's team calls their final witness, Isaac Barrow (played enthusiastically by Bernice).

Newton's attorney: "How did you meet Sir Isaac Newton?"

Barrow: "It was 1667, in an optics lecture I was giving. He was a geometry student. It was in the early 1670s when I saw his work on calculus. He took a lot of scorn and criticism. He was concerned about his work, since it was not concrete like geometry was."

Newton's attorney: "As a professional thinker, if you had ever published a paper in which you'd made use of another's work, would you acknowledge the other in your work?"

Barrow: "Yes. To not do so would be plagiarism."

Newton's attorney: "Knowing Newton as you do, would Newton have accused Leibniz of plagiarism if he'd not been guilty of it, if he'd merely collaborated with Newton's full knowledge of it?"

Barrow: "No, and he would have been happy with Leibniz if he'd given credit where credit was due."

Newton's attorney: "Is it possible that Leibniz used Newton's work as a stepping stone to complete his own?"

Barrow: "Yes."

Newton's attorney: "And Newton would have been all right with this had Leibniz given him credit?"

Barrow: "Yes, there'd be no argument."

The defense had no questions for Barrow.

10:02 -- 10:04. Leibniz's team offers their closing argument: "Both of these men invented calculus, and invented it by different means. As a remark, note that the two men never met face-to-face, never collaborated, and neither likely knew what the other looked like. I would also like to point out that Newton used fluxions and fluids and not infinitesimals, and we have never accused Newton of stealing the ideas of another."

10:04 -- 10:06. Newton's team offers their closing statement: "Sir Isaac Newton, President of the Royal Society. Undeniably both of these men have great minds, but the point of this trial was to establish whether ot not Gottfried Leibniz stole Newton's work. Newton began his work prior to Leibniz's beginning his own work, and this work was available to Leibniz, so it could have been done. Leibniz saw Newton's manuscripts, and there's no way he could have ignored what he'd seen."

***

The things I like best about this re-enactment:

1. The students' robust preparation: every single one actively involved in the court proceedings showed they'd practiced their roles carefully and had mastered the material they'd be asked to discuss.

2. The students' arguments: while the Leibniz team did all they could to drive a wedge between the two scholars' methods (no doubt hoping the jury would be convinced they were different enough to have been, undeniably, developed independently), Newton's attorneys focused on the character of their client, arguing (more subtly at times than at others) that he was a trustworthy man, honest and upstanding, surely incapable of making false accusations of plagiarism.

3. The students' performances: with only one or two (understandable!) exceptions, none of the students broke character, and every one offered sterling deliveries.

Coming up next: in their own words, students' reactions to the activity, and a little commentary of my own.

Here's a shot of the Venn diagram we made for yesterday's exercise (my apologies for the crappy quality of my cell phone's camera):

As we decided in class, the sets of pure, radical, finite/algebraic, and arbitrary extensions make up a rough map of the inner solar system, while the normal field extensions act as a short-period comet orbiting the rational sun on an elliptical orbit pointing to the upper left corner of the picture. (The field of complex numbers, clearly visible in the upper left, marks the comet's aphelion.)

Friday, April 24, 2009

Well...perfect might be a stretch, but it was a good one, capping off a great week. The jewel in the crown of today's classroom antics shone brightly in the afternoon sun.

As soon as I reached campus at 8:30 I began fielding questions regarding the second of two homework problems I'd assigned to my Abstract II students for today; Quincy caught me as I walked in the door to the department with a couple of questions. A few hours later he was at it again in the Math Lab, and Derrick, Nadia, Hermann, Opal, and Katya all had difficulties too. Clearly the question (to which I openly admitted I knew not all the answers) was proving rougher than I'd thought it would.

The problem is simple to state: starting with the field of rationals, describe the various intercontainments of the following sorts of extensions that field: all extensions, finite extensions, algebraic extensions, normal extensions, pure extensions, and radical extensions. (Each containment or failure of containment had to be justified by proof or example.)

Between five or six of us, by 2:30 (class begins at 2:45) we'd catalogued examples and proofs verifying all but a couple of the containments/noncontainments we needed to show, but it was clear that the best course of action was to bring it up together in class.

That we did, and our team solution was an exciting one.

Several of the containments were easy and indisputable: pure extensions are radical, radical extensions are finite, finite extensions and algebraic extensions are one and the same. What wasn't clear was how normal extensions fit into the picture. It took us about half an hour of field wrangling before we'd found all of the examples we'd needed, constructing a convoluted splitting field here, a two-step radical tower there.

We were done.

Or so most of us thought.

Miguel had the temerity to ask about the exact location of the complex numbers in the scheme we'd laid out.

After a Moonlightingesque (kids, ask your parents) three minutes of high-level mathematical crosstalk, we remembered that the equivalence of two characterizations of normal extensions only holds for finite extensions (of which the complex numbers are not one), and therefore we were definitely allowed to call the complex numbers a normal extension of the rationals.

I realize that much of the above discussion will be over the heads of uninitiated, but I wanted to provide something close to a blow-by-blow account in order to highlight the phenomenal work of several of my students. Derrick, Nadia, Quincy, Miguel, Hermann, and Bertrand all contributed substantially to the conversation. The result was a true work of social mathematics, a finely-crafted piece of collaborative art.

Moreover, through our class today the students and I all gained a better understanding of the interrelationships of the manifold definitions we've encountered over the past few weeks; I can think of few fifty-minute periods this past semester that have been more mathematically or pedagogically satisfying. Though the exercise was definitely more challenging than I had intended it to be, it proved to be not insurmountably challenging, and the result of the solution was to solidify our fundamental understanding of several tricky concepts. This exercise is definitely a keeper, and I may just use it as a classroom activity that next time I teach this course (however long from now that will be).

I wonder if I might find similar exercises for my upcoming classes next fall? Or for the REU students this summer as they undertake a brutal and bruising survey of graph theory in the second week of June...?

Thursday, April 23, 2009

It's been a great week. Now that I'm over the semester's hump (for me the period of complete and utter insane busyness ended about a week ago), I'm relaxed enough to enjoy the day-to-day interaction with my students.

There's nothing like a Thursday for that kind of deal, and today was a wonderful one: I had several awesome meetings with students. A few just wanted me to vet homework solutions (way to go, Karlie! Your answer was spot on! Gorgeous!), others had meetings with me to talk about their upcoming presentations in 280 or 462 (Pascal's fractal and Ramsey numbers! This stuff is awesome, and your presentations are gonna rock!), and another met with me so we could talk turkey about undergraduate research into poset representation of graph coloring games. (Tish, you are so going to clean up on this one...)

I've been in a good mood all week. It doesn't hurt that we're doing awesome stuff in all three of my classes: Riemann sums in Calc I, sizes of infinity in 280, and Galois groups in 462...what's not to like?

It's cheesed me off a little bit that it's taken me roughly 13 weeks to really get into the groove with this semester...and now it's almost over.

Well, I've nearly managed to survive yet another term, and I don't mind saying this one's been brutal. The four-prep schedule (one new, one a rerun but with a new text, and a third, also a rerun, but with a substantial new component) has been a bear: in some weeks it took all that I had just to take care of the bare essentials for each of my classes. I hope my weariness has not shown itself too starkly.

Ah, well. Water under the bridge, I trust.

For now, let's enjoy the sunshine and the last week or so we've got together. This goes especially to my graduating seniors, with some of whom I've enjoyed as many as six classes over the past four years. Nadia and Sylvester top that list...I'm going to miss you guys! It won't be the same department without either of you around.

Tuesday, April 21, 2009

1. Your days of ten-minute, plug 'n' chug, follow-the-formula homework problems are over: these problems will screw your brain 'til it hurts and haunt you as you lie in bed at night.

2. There's no better reason to start your homework as soon as it's assigned than the fact that those who are likely to complete it in quality fashion will spend 10 to 15 hours on it before they submit their final drafts.

3. Even if you've cruised through your math classes in the past by putting forth minimal effort, you're almost certainly going to have to work, and work hard, to make it through this one.

4. Sure, you may be working your tail off, but please know that ultimately there's no substitute for good, honest struggle: true learning is almost invariably accompanied by trial, error, and hair-pulling frustration.

5. You're not the only one working your tail off.

Related note: I already know what one of the 280 students' final exam questions is going to be...

Monday, April 20, 2009

I've got some random ruminations to share, in the wake of this past Saturday's lovely WAC/CAC (that's Writing Across the Curriculum and Communication Across the Curriculum, for those not in the know) Symposium at UNC Greensboro.

Please discuss, in the comments section, anonymously, if you'd like:

1. Why are faculty so fearful of assessment? Is it because of the work involved? Is it that they're just not sure how best to do it, or are they afraid of what they'll find if they do it right?

2. For those of you out there who are into math (faculty especially): when in life was it that you first fell in love with math? What is it about mathematics that drew you in, that got you hooked?

3. Are my homework committees working as well as they ought to? Is there a different model I might adopt that would make them more effective? (For what it's worth, I think they work more effectively in 280 than they do in the 400-level courses; while the 280 students don't often "get the right answer" from their committee-dwelling peers, they're exposed to multiple points of view on the same problem, which is ultimately far more important than getting the right answer anyway.)

4. For my students: just how busy are you? I get the sense that this particular Spring 2009 semester has been one of the most stressful on record, for faculty and students alike, and I suppose a lot of it has to do with the economic situation the world's found itself in, and the commensurate busyness it's forced on us all as we struggle to make ends meet by asking 1 plus 1 to be 3.

Monday, April 13, 2009

It took half of my waking hours today to go through my students' reflections on the Newton v. Leibniz project, respond to them (with over 5000 words of my own), and excerpt them for my own nefarious ends (a CoB coming soon!), but it was well worth it.

I think this iteration of the project succeeded in impressing on the students the importance not only of the mathematics itself but also the way in which it was discovered/invented. The students' reflections were almost universally mature, well-thought-out, well-composed, and insightful. While a few offered ways in which the project itself could be improved, almost everyone was delighted with its effectiveness, and many lauded it as one of the best learning experiences they've ever had.

Huh. Zah.

As promised obliquely above, a deeper discussion is soon to come, as well as a blow-by-blow account of the trial itself. But now, if you'll excuse me, bed awaits: this weekend's offered up another maelstrom of grading (six different assignments, totaling roughly 13 hours), and I've only seven hours before tomorrow's day commences.

Monday, April 06, 2009

Easily navigated, centrally located venueFive to ten (5-10) dedicated facultyCommitted students, proportional to facultyAnimated and understandable plenary speakers (one to three, time permitting)Inexpensive food and unpresuming (pizza, potato chips, potato soup)Hospitably hosted game nightStudent presentations (with reasonable expectations)Willingness to let the students run the show

Begin with a generous serving of inexpensive and unpresuming food, mix in the bulk of the students and faculty and add a plenary speaker. Cover with a game night and more inexpensive food, and let sit overnight.

Come morning, begin with a warm welcome from a dedicated faculty member and another plenary speaker. Add student talks, a few at a time, making sure the students control the content and pace of their own talks and oversee the talks’ timing and question-and-answer periods. Be sure not to overheat. Cover with another serving of unpresuming food. Repeat this layering process two or three times.

Coat the top with generous closing remarks from one of the dedicated faculty (door prizes optional).